weakening(?) of locally connected

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

As is well known a geometric morphisms F --| U : EE -> SS is locally
connected iff F preserves dependent products.
But has anyone come across a g.m. whose inverse image part preserves
ordinary function spaces, i.e. exponentials, but not dependent
function spaces?

In connection with Lawvere and Menni's TAC 30 paper (section 10) this
question has been asked under the additional assumption that the
g.m. is also hyperconnected and local.

Triggered by discussion with Matias Menni I have thought about it with
moderate success and then found out that I even do not know the answer
to the (possibly) simpler question formulated at the beginning of this
message.
My hope now is that someone happens to know a counterexample to this
simpler question at least...

In my eyes the requirement that the further left adjoint be
fibered/indexed is most natural if one thinks of gm's as cocomplete
and locally small toposes over an arbitrary base topos (which I do as
is apparent from my notes on Fibered Categories (section 18)).

All examples of precohesive gm's I know of are locally connected. But
it would be very surprising if this were always the case (unless the
base is Set where essential and locally connected are known to coincide).

Thanks in advance,

Thomas





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